reserve a,b,c for set;

theorem Th11:
  for x,y being Real holds [.x,y.[ is open Subset of Sorgenfrey-line
proof
  let x,y be Real;
  reconsider V = [.x,y.[ as Subset of Sorgenfrey-line by Def2;
  now
    let p be Point of Sorgenfrey-line;
    reconsider a = p as Element of REAL by Def2;
    assume
A1: p in [.x,y.[;
    then
A2: x <= a by XXREAL_1:3;
    a < y by A1,XXREAL_1:3;
    then consider q being Rational such that
A3: a < q and
A4: q < y by RAT_1:7;
    reconsider U = [.x,q.[ as Subset of Sorgenfrey-line by Def2;
    take U;
    x < q by A2,A3,XXREAL_0:2;
    hence U in BB by Lm5;
    thus p in U by A2,A3,XXREAL_1:3;
    thus U c= V by A4,XXREAL_1:38;
  end;
  hence thesis by Lm6,YELLOW_9:31;
end;
